5.4 Isotropy: Hamiltonian constraint

Dynamics is controlled by the Hamiltonian constraint, which classically gives the Friedmann equation.
Since the classical expression (29) contains the connection component , we have to use holonomy
operators. In quantum algebra we have only almost-periodic functions at our disposal, which does not
include polynomials such as . Quantum expressions can, therefore, only coincide with the classical one
in appropriate limits, which in isotropic cosmology is realized for small extrinsic curvature, i.e., small in
the flat case. We thus need an almost periodic function of , which for small approaches . This
can easily be found, e.g., the function . Again, the procedure is not unique since there are many such
possibilities, e.g., , and more quantization ambiguities ensue. In contrast to the
density , where we also used holonomies in the reformulation, the expressions are not
equivalent to each other classically, but only in the small-curvature regime. As we will discuss
shortly, the resulting new terms have the interpretation of higher-order corrections to the classical
Hamiltonian.

One can restrict the ambiguities to some degree by modeling the expression on that of the full theory.
This means that one does not simply replace by an almost periodic function, but uses holonomies
tracing out closed loops formed by symmetry generators [50, 54]. Moreover, the procedure can be
embedded in a general scheme that encompasses different models and the full theory [292, 50, 68], further
reducing ambiguities. In particular models with non-zero intrinsic curvature on their symmetry orbits, such
as the closed isotropic model, can then be included in the construction. (There are different
ways to do this consistently, with essentially identical results; compare [111] with [28] for the
closed model and [300] with [285] for .) One issue to keep in mind is the fact that
“holonomies” are treated differently in models and the full theory. In the latter case they are ordinary
holonomies along edges, which can be shrunk and then approximate connection components. In
models, on the other hand, one sometimes uses direct exponentials of connection components
without integration. In such a case, connection components are reproduced in the corrections
only when they are small. Alternatively, a scale-dependent can provide the suppression
even if connection components remain large in semiclassical regimes. The requirement that this
happens in an acceptable way provides restrictions on refinement models, especially if one goes
beyond isotropy. Selecting refinement models, on the other hand, leads to important feedback
for the full theory. The difference between the two ways of dealing with holonomies can be
understood in inhomogeneous models, where they are both realized for different connection
components.

In the flat case the construction is easiest, related to the Abelian nature of the symmetry group. One
can directly use the exponentials in (44), viewed as 3-dimensional holonomies along integral
curves, and mimic the full constraint where one follows a loop to get curvature components of
the connection . Respecting the symmetry, this can be done in the model with a square
loop in two independent directions and . This yields the product , which
appears in a trace as in Equation (15), together with a commutator , using the
remaining direction . The latter, following the general scheme of the full theory reviewed in
Section 3.6, quantizes the contribution to the constraint, instead of directly using the simpler
.

Taking the trace one obtains a diagonal operator

in terms of the volume operator, as well as the multiplication operator

as the only term resulting from in . In the triad representation,
where instead of working with functions one works with the coefficients in an
expansion , this operator is the square of a difference operator. The constraint equation
thus takes the form of a difference equation

for the wave function , which can be viewed as an evolution equation in internal time . Thus,
discrete spatial geometry implies a discrete internal time [51]. The equation above results in the
most direct path from a non-symmetric constraint operator with gravitational part acting as

Operators of this form are derived in [54, 16] for a spatially-flat model (), in [111, 28, 286] for
positive spatial curvature (), and in [300, 285] for negative spatial curvature (), which is
not included in the forms of Equations (53) and (54). Note, however, that not all these articles used the
same quantization scheme and thus operators even for one model appear different in details, although the
main properties are the same. Some of the differences of quantization schemes will be discussed below. The
main option in alternative quantizations relates to a possible scale dependence , which we leave open
at this stage.

One can symmetrize this operator and obtain a difference equation with different coefficients, which we
do here after multiplying the operator with for reasons that will be discussed in the context of
singularities in Section 5.16. The resulting difference equation is

where .

Since , the difference equation is of higher order, even formulated on
an uncountable set, and thus has many independent solutions if is constant. Most of them, however,
oscillate on small scales, i.e., between and with small integer . Others oscillate only on
larger scales and can be viewed as approximating continuum solutions. For non-constant , we have a
difference equation of non-constant step size, where it is more complicated to analyze the general form of
solutions. (In isotropic models, however, such equations can always be transformed to equidistant form up
to factor ordering [75].) As there are quantization choices, the behavior of all the solutions leads to
possibilities for selection criteria of different versions of the constraint. Most importantly, one chooses the
routing of edges to construct the square holonomy, again the spin of a representation to take the
trace [170, 299], and factor-ordering choices between quantizations of and . All these
choices also appear in the full theory, such that one can draw conclusions for preferred cases
there.